We construct Lorentz invariant and gauge invariant 1PI effective action for closed and open superstrings and demonstrate that it satisfies the classical BV master equation. We also construct the quantum master action for this theory satisfying the quantum BV master equation and generalize the construction to unoriented theories. The extra free field needed for the construction of closed superstring field theory plays a crucial role in coupling the closed strings to D-branes and orientifold planes.

We propose a toy model for holographic duality. The model is constructed by embedding a stack of NN D2-branes and KK D4-branes (with one dimensional intersection) in a 6d topological string theory. The world-volume theory on the D2-branes (resp. D4-branes) is 2d BF theory (resp. 4D Chern-Simons theory) with \mathrm{GL}_NGLN (resp. \mathrm{GL}_KGLK) gauge group. We propose that in the large NN limit the BF theory on \mathbb{R}^2ℝ2 is dual to the closed string theory on \mathbb{R}^2 \times \mathbb{R}_+ \times S^3ℝ2×ℝ+×S3 with the Chern-Simons defect on \mathbb{R} \times \mathbb{R}_+ \times S^2ℝ×ℝ+×S2. As a check for the duality we compute the operator algebra in the BF theory, along the D2-D4 intersection – the algebra is the Yangian of \mathfrak{gl}_K𝔤𝔩K. We then compute the same algebra, in the guise of a scattering algebra, using Witten diagrams in the Chern-Simons theory. Our computations of the algebras are exact (valid at all loops). Finally, we propose a physical string theory construction of this duality using D3-D5 brane configuration in type IIB – using supersymmetric twist and \OmegaΩ-deformation.

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.